220 XIII INTEGRATION (1) Uniqueness. Every compact subset of X x Y is contained in one of the form L x M, where L and M are relatively compact open subsets of X and Y respectively ((3.20.17) and (3.18.2)). We shall show that, for each function h e 3tfc(X x Y) with support contained in L x M, the value of v(h) is well-determined. This will follow from (1 3.21 .1 .2) If L c X and M c Y are relatively compact open sets, then every function h e JTC(X x Y) with support contained in L x M is a cluster point, in the Banach space Jf*c(X x Y; L x M), of the set of functions of the form where (/,) is a finite sequence of functions in Jfc(X) (xyy)^>^fi(x)gk(y\ where (/,) is a finite sequence o unctions in JcX i,k with support contained in L, and(gk) is a finite sequence of functions in «?f C(Y) with support contained in M. Assuming this for the moment, by hypothesis there exists a number c> 0 such that | v(w)| ^ c * \\u\\ for all u e Jfc(X x Y) with support contained in L x M. On the other hand, for each s > 0 there exist two finite sequences of functions /£ e tfc(X)9gk e Jfc(Y) such that for all (x, y) e L x M, the support of the left-hand side being contained in L x M. Using (13.21.1.1), it follows that i, k and since s is arbitrary, this establishes the uniqueness of v. To prove (13.21.1.2), let e be a positive real number. Then for each z = (X y) e L x M there exists a compact neighborhood U c L (resp. V c M) of x in X (resp. of y in Y) such that the oscillation of h in U x V is :ge (3.20.1). The projections S c L and T c= M of Supp(/z) are compact, hence for each x e S there exists a finite number of points yfa) (1 ^y ^ n(x)) of T, and for eachy a compact neighborhood U/x) cz L of x in X and a compact neighborhood V/.y/X)) c M of y^(x) in"Y, such that the oscillation of h in Vj(x) x Vj(yj(x)) is g e, and such that the interiors of the sets Vj(yj(x)) cover T. The set W(x) = Q U/(x) is a compact neighborhood of x in X. Hence there exists a finite number of points xt (1 ^ i g m) in S such that the interiors of the sets U'(.Xf) cover S. Put Af = U'(*i), and let (Bk)i^k^P be the family of sets obtained as follows: for each point y E T, let W(y) be the inter- section of the (finitely many) interiors of sets VjO^jCj)) which contain y; the sets WOO are °Pen and nonempty, and form a finite open covering of T which we denote by (Bfc)^fc^p. Notice also that, by construction, the oscillation of h in each set A{ x Bfe is ^ e. Now let (/^ ^ f^w (resp. (g^)l ^k^p) be continuous mappings of X (resp. Y) into [0, 1], such that Supp(/f) c A4 and Supp(^) c Bfclowing